Pompeiu's problem on discrete space.

Area of Science:

• Harmonic Analysis
• Geometric Measure Theory
• Multidimensional Analysis

Background:

• The Pompeiu problem traditionally examines unique function properties based on geometric shapes.
• Extending this problem to higher dimensions and families of sets presents unique mathematical challenges.
• Understanding function behavior on lattices is crucial in various fields, including signal processing and physics.

Purpose of the Study:

• To generalize the Pompeiu problem to finite families of finite subsets within the n-dimensional lattice Z(n).
• To identify the precise conditions on these families of subsets that force a function to be identically zero.
• To analyze the implications of translation invariance within the lattice group tau.

Main Methods:

• The study employs techniques from Fourier analysis and spectral theory.
• It involves analyzing the properties of specific integral operators defined over the lattice Z(n).
• Methods include investigating the null spaces of these operators under varying family configurations.

Main Results:

• Characterization of families of subsets for which the Pompeiu problem has only the trivial solution.
• Identification of key geometric or combinatorial properties of the subsets that dictate the solution.
• The results provide necessary and sufficient conditions for the zero function to be the unique solution.

Conclusions:

• The Pompeiu problem on n-dimensional lattices is solvable and depends critically on the geometric configuration of the subset family.
• The findings contribute to the understanding of unique continuation properties for functions on discrete spaces.
• This research offers a foundation for further exploration of integral equations and function analysis in higher-dimensional lattices.